These mathematical expressions allow us to know the relationship between variables.
This article aims to be a compilation of the different types of mathematical functions that exist, explaining them in greater detail and discovering what is the formula that allows us to operate.
What are mathematical functions?
Mathematical functions are defined as the mathematical expressions existing between two variables. These variables, or magnitudes, are represented using the letters “x” and “y”, forming the domain and codomain respectively.
The relationship between these two components tries to find equality between domain and codominium. Generally, for each value of the “x” there is a single value for “y”, although not all mathematical functions meet this requirement.
In addition to allowing us to know the relationship between two variables, the functions allow you to represent it graphically that exposes much more clearly how one variable is going to behave in function of the other.
The 15 types of mathematical functions
Below we show several types of functions classified into two categories: algebraic functions and transcendent functions.
This classification is made with functions in which the value of the domain or “x” corresponds to a single value in the codomain or “y”.
1. Algebraic functions
They are functions that are characterized by establishing a relationship between their components through monomials or polynomials and using basic mathematical calculations: addition, subtraction, multiplication, division and use of roots.
Within the algebraic functions we can find the following subtypes, we see them below.
1.1. explicit functions
They are those functions in which it is possible to obtain the relationship between “x” and “y” in a relatively simple way, substituting the value of “x” and seeing how it influences the other variable.
The implicit function is one that is given in the form that the variable “y” is clear. An example of such functions:
y=3x
1.2. Implicit functions
They differ from explicit functions in that the relationship between domain and codomain is not established directly. It is necessary to do several transformations in order to find out how the “x” and “y” are related. In implicit functions, the variable “y” is not clear.
y+3x=9
1.3. polynomial functions
Polynomial functions are those whose expression is given in the form of a polynomial, that is, a more or less extensive numerical expression.
There are different types, being classified according to their degree.
Among the polynomial functions of the first degree we find the affine functions, the linear ones and the identity ones.
Affine functions are those that, when represented graphically, the line does not pass through the origin of the coordinates, that is, the point (0,0). These lines are defined according to the following formula:
f(x)=mx+n
Linear functions, unlike affine ones, do pass through the origin of the coordinates. Its formula is the same as in related formulas, only without the “n”.
f(x)=mx
Identity functions are functions whose slope “m” has the value 1. As in linear functions, they pass through the point (0,0) and divide the first and third quadrants into equal parts:
f(x)=x
Quadratic functions fall into the category of second-degree polynomial functions, so called because the largest exponent of the polynomial is 2, and its graphic representation is that of a vertical parabola:
f(x)=ax^2+bx+c
Finally, cubic functions are polynomial functions of the third degree because their greatest exponent is 3, being represented in the form of an S rotated 180º. An example of a cubic function would be:
f(x)=ax^3+bx^2+cx+d
1.4. rational functions
They are those functions in which their value is established from a quotient between two non-zero polynomials. These functions include all numbers except the one that makes y=0 happen.
In this type of functions there are limits, which are called asymptotes, and they come to indicate that it is the value that makes the function end up being zero as a result.
The graphical representations in this type of functions have a tendency to go to infinity, avoiding touching the lines that form the asymptotes.
1.5. irrational functions
They are rational functions within a radical. These roots impose certain restrictions when solving the function, such as the fact that the values of “x” must make the result always give positive values.
y = √f(x)
1.6. piecewise functions
In these functions there are different formulas depending on the piece represented in the graph. There can be three formulas and, depending on the value of “x”, use one or the other.
2. Transcendent functions
They are those functions that are performed when, to obtain the relationships between two variables, it is not possible to perform algebraic functions. These involve having to carry out a complex calculation process in order to discern the relationship between the magnitudes.
Being more complex than algebraic functions, transcendental functions make use of derivatives, integrals, and algorithms.
2.1. exponential functions
They are those functions in which the relationship between the domain and the codomain is exponential. Graphically, the relationship would be exposed in the form of a line that grows continuously, whose growth would become increasingly larger and more inclined.
In these functions, the “x” is the exponent value.
f(x)=a^x
2.2. logarithmic functions
Unlike exponential functions, as the function evolves, the growth is reduced more and more.
In these functions, the value of “x” must be greater than zero and less than 1.
y= log x
23. Trigonometric functions
These types of functions allow us to know the relationships between the elements that make up a geometric figure, especially triangles.
Within the trigonometric functions are the calculation of sine, cosine and tangent. which can be very useful to calculate the height of mountains and buildings.
other functions
We are going to learn about other types of functions that cannot be included in the previous categories.
1. Injective functions
A function is injective if each element of “y” has a single element in the form of an “x” to which it corresponds. That is, there cannot be more than one value “x” for the same value of “y”.
2. Surgeese functions
They are those functions in which all the values for “y” are related, at least, with one of “x”, although it does not necessarily have to be linked only and exclusively with one.
3. Bijective functions
They are those functions in which both injective and surjective properties are given. That is, there is a single value of “x” for “y”, and all values in the domain correspond to one in the codomain.
4. Non-injective and non-surjective functions
In these functions there are several domain values for a codomain, or what is the same, that there can be several values for “x” that give the same of “y”.
References
- Eves, H. (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover.
- Hazewinkel, M. ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers.
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